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A new perspective on the aerodynamic performance and power limit of crosswind kite systems

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A new perspective on the aerodynamic performance and power limit of crosswind kite systems

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In this paper, a new perspective on the aerodynamic performance modelling of crosswind kite power systems (CKPSs) is provided, where the effects of the induction factor or flow retardation by the kite are taken into account. For simplicity, only CKPSs in straight downwind configuration are considered, where the kites sweep an area perpendicular to the wind direction. Moreover, the in-plane or tangential induction factor is neglected. It is argued that the concept of the swept-area-normalized power coefficient, which is commonly used for conventional wind turbines, is not practically important for CKPSs. Instead, a kite-area-normalized power coefficient concept is adopted and is shown to be a more appropriate metric of performance of kite systems. The kite-area-normalized useful and wasted power coefficients for both a lift and a drag mode CKPS are plotted at different values of solidity factor and aerodynamic efficiency. Moreover, it is shown that the two modes of power generation, i.e. lift and drag modes, yield the same amount of useful power when the kite system solidity factor is essentially zero – this is in agreement with the underlying assumption in the crosswind kite power principle. For non-zero solidity factors, thus non-zero induction factors, however, our results suggest that the drag mode has higher power generation potential than the lift mode.
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A new perspective on the aerodynamic performance and power1
limit of crosswind kite systems2
Mojtaba Kheiria,b,
, Vahid Saberi Nasrabadb, Fr´ed´eric Bourgaultb
3
aConcordia University, 1455 de Maisonneuve Blvd. West, Montr´eal, QC, H3G 1M8, Canada4
bNew Leaf Management Ltd., 500-1177 West Hastings Street, Vancouver, BC, V6E 2K3, Canada5
Abstract6
In this paper, a new perspective on the aerodynamic performance modelling of crosswind
kite power systems (CKPSs) is provided, where the effects of the induction factor or flow
retardation by the kite are taken into account. For simplicity, only CKPSs in straight
downwind configuration are considered, where the kites sweep an area perpendicular to the
wind direction. Moreover, the in-plane or tangential induction factor is neglected. It is argued
that the concept of the swept-area-normalized power coefficient, which is commonly used
for conventional wind turbines, is not practically important for CKPSs. Instead, a kite-area-
normalized power coefficient concept is adopted and is shown to be a more appropriate metric
of performance of kite systems. The kite-area-normalized useful and wasted power coefficients
for both a lift and a drag mode CKPS are plotted at different values of solidity factor and
aerodynamic efficiency. Moreover, it is shown that the two modes of power generation, i.e. lift
and drag modes, yield the same amount of useful power when the kite system solidity factor
is essentially zero – this is in agreement with the underlying assumption in the crosswind kite
power principle. For non-zero solidity factors, thus non-zero induction factors, however, our
results suggest that the drag mode has higher power generation potential than the lift mode.
Keywords: Airborne wind energy, crosswind kite power system, induction factor,7
Betz-Joukowsky limit, actuator disc theory, crosswind principle8
Corresponding author. Tel.: +1 514 848 2424 ext. 4210.
Email address: mojtaba.kheiri@concordia.ca (Mojtaba Kheiri)
Preprint submitted to Journal of Wind Engineering & Industrial Aerodynamics July 22, 2019
1. Introduction9
Crosswind kite power systems (CKPSs) are a type of airborne wind energy (AWE) systems
10
which are used for harnessing high-altitude wind energy. Winds at higher altitudes are stronger
11
and steadier, thus greatly appealing for power generation. The principle of “crosswind kite
12
power” was first introduced in a seminal paper by Loyd (1980), who showed that large
13
amounts of wind power could be harvested inexpensively by means of an aerodynamically
14
efficient tethered wing (or kite) flying at high speed transverse to the incoming wind direction.
15
He also proposed two different modes for power generation, namely the lift mode (i.e. ground-
16
based generation) and the drag mode (i.e. on-board generation); see Figures 1and 2. In
17
the lift mode, power is generated by transferring mechanical power via tether tension to
18
the ground (e.g. unrolling the tether from a drum), while in the drag mode, electricity is
19
generated by on-board turbines and transmitted via conductive material in the tether to
20
the ground. Various concepts employing, e.g. fixed wings, soft kites, autogyros, or airships,
21
have so far been put forward to extract high-altitude wind energy; for more details about
22
these concepts, the interested reader is referred to Fagiano and Milanese (2012); Ahrens et al.
23
(2013); Cherubini et al. (2015); Rancourt et al. (2016); Schmehl (2018).24
Nearly three decades after introduction of the crosswind kite power principle, Argatov
25
et al. (2009) proposed a “refined crosswind motion law” for lift mode kites, including the
26
effects of the tether drag. They also obtained a new expression for the peak value of the
27
mechanical power, taking into account the effects of centrifugal, gravitational and frictional
28
forces. In a subsequent paper, Argatov et al. (2011) extended the refined crosswind motion
29
law to include the effects of kite’s control and gravity.30
To date, most studies on the aerodynamic performance of CKPSs have neglected the
31
effects of flow retardation or induction factor by a kite. The induction factor is a measure
32
of the influence of an energy harvesting device (e.g. a wind turbine) on flow, and it may
33
2
be correlated to the capability of the device to harvest power from flow. There have been,
34
nevertheless, a few attempts to include the effects of the induction factor in the aerodynamic
35
modelling of CKPSs. For example, Zanon et al. (2014) explored ‘the impact of the airfoil-
36
airmass interaction on the extracted power’ of a single- and dual-airfoil drag mode systems.
37
They showed that, for a single large-scale kite (
14MW), the extracted power is reduced by
38
27% , if in the power calculations, the airfoil-airmass interaction is also taken into account.
39
Recently, Leuthold et al. (2017) examined the relevance of axial (i.e. normal to the rotor
40
plane) and lateral (i.e. in-plane) induction factors to the modelling of lift mode multiple-kite
41
airborne wind energy systems (MAWESs). They found that the axial induction factor is
42
relevant and should be considered in the power optimization of a MAWES, while the lateral
43
induction factor may be neglected.44
Recently, Kheiri et al. (2017a,2018) developed, in a systematic manner, an extended
45
actuator disc theory that included the effects of the induction factor. This theory was then
46
used to derive expressions for the potential power output from a kite in both lift and drag
47
modes. They also proposed a formula for calculating the induction factor of a CKPS. Their
48
numerical results showed that the induction factor for a crosswind kite system may, in fact,
49
be quite significant, depending on the system parameters, in contrast to what is commonly
50
assumed. In addition, it was found that neglecting even a small induction factor may result in
51
a significant overestimation of power output. Moreover, Kheiri et al. (2018) conducted several
52
CFD simulations to validate their theory. The CFD results showed a good agreement with
53
the theoretical results. In addition, the CFD simulations showed that a low-speed turbulent
54
wake flow is formed and expands to distances several times of the gyration radius behind a
55
CKPS, quite similarly to conventional wind turbines (CWTs). Studying the wake flow of a
56
single or multiple kites is essential for kite farms layout design and optimization; for some
57
details see Kheiri et al. (2017b).58
3
It is well-known that an energy extracting device, such as a wind turbine, cannot theoret-
59
ically harvest 100% of the available wind power. The amount of power than can be extracted
60
from a freestream via an energy extracting device is theoretically limited to 16/27 (or
59%)
61
of the available wind power. This is commonly referred to as the Betz-Joukowsky (B-J, in
62
short) limit and may be derived from the actuator disc theory (see Okulov and van Kuik,
63
2009). Several investigators have expressed reservations about the applicability of the B-J
64
limit to crosswind kite systems; for example, refer to Loyd (1980); Archer (2013); Costello
65
et al. (2015); alternatively, the reader may refer to Kheiri et al. (2018) for a brief discussion
66
of the issue.67
Nonetheless, in a very recent paper, De Lellis et al, (2018) attempted to extend the
68
concept of the B-J limit to CKPSs. Their parallel development has resulted in the same
69
expressions as those presented in Kheiri et al. (2017a,2018) for the power output of lift
70
mode CKPSs.
1
They have argued that a horizontal-axis wind turbine (HAWT) and a drag
71
mode kite system extract power through similar means (i.e. torque in HAWT and thrust
72
of on-board turbines in drag mode CKPS). They concluded then, similarly to HAWTs, the
73
classical B-J limit should apply to a drag mode CKPS. They also showed that a lift mode
74
CKPS at best can harvest 4/27
15% of the available wind power – that is 1/4 of the B-J
75
limit. In addition, they found that a lift mode CKPS at the optimal point should spend
76
exactly the same amount of power as the useful power to drive the kite.77
The present paper aims to provide a new theoretical perspective on the aerodynamic
78
modelling and power limit of CKPSs by taking into account the effects of the induction factor.
79
For simplicity, only CKPSs in straight downwind configuration are considered here, where the
80
kites sweep an area perpendicular to the wind direction while also reeling out (i.e. translating
81
1
This remarkable coincidence, which should perhaps not come as a surprise, raises our level of confidence
in the results to be correctly representative of the physics behind the system.
4
downwind) at a constant speed. Two different but relevant definitions of the power coefficient,
82
i.e. swept- and kite-area-normalized power coefficients, are provided and their applicability
83
to CKPSs will be discussed. Using the formula for obtaining the induction factor, proposed
84
by Kheiri et al. (2017a,2018), the kite-area-normalized power coefficients (i.e. both useful
85
and wasted portions) are expressed as functions of independent dimensionless variables, such
86
as the solidity factor, and aerodynamic efficiency (refer to Section 3for definitions).87
The paper is organized as follows: first, a few important outcomes of the extended actuator
88
disc theory are described in Section 2to serve as background reference to the rest of the paper.
89
In Section 3, expressions for the kite-area-normalized useful and wasted power coefficients for
90
lift mode CKPSs are presented, and their variations as a function of the independent variables
91
(solidity factor and aerodynamic efficiency) are illustrated. In Section 4, the corresponding
92
power coefficient expressions for drag mode CKPSs are presented and plotted. Finally, in
93
Section 5, the useful power harvesting (or power output) potential of a drag mode CKPS
94
is compared to that of a lift mode. It will be argued that a drag mode CKPS may have a
95
greater potential for useful power harvesting compared to a lift mode CKPS with the same
96
design parameters.97
2. Background98
Figure 3shows a wind turbine rotor subjected to a uniform wind field of absolute velocity
99
v
and moving at constant speed
vd
in the wind direction (i.e. downwind). By defining
e
as
100
the ratio of reel-out speed to freestream velocity,
vd
may be linked to
v
as
vd
=
ev
, where
101
0
e
1. Following Sørensen (2016); Kheiri et al. (2017a,2018); De Lellis et al, (2018), the
102
induction factor for a moving actuator disc or simply a rotor may be defined in connection to
103
the relative incoming flow velocity/wind speed, (vvd), that is104
vr(vvd)(1 a),(1)105
5
in which vris the actual relative flow velocity at the rotor, and ais the induction factor.106
It follows from the extended actuator disc theory, developed for a moving disc (refer to
107
Kheiri et al.,2018), that the absolute wake velocity is dependent on both aand e:108
vw=v[1 2a(1 e)].(2)109
Since a negative relative wake velocity, i.e.
vwvd
, meaning a reverse wake flow into the
110
moving control volume, is not sensible for an energy harvesting device, the acceptable range
111
for ais obtained by letting vwvd0 which eventually leads to:112
0a1
2.(3)113
On the other hand, the axial load
2
or thrust, denoted by
T
, acting on the disc or rotor and
114
the extracted power by the rotor, P, may be written as115
T=1
2ρAsv2
4a(1 a)(1 e)2,(4)116
and117
P=T vr=1
2ρAsv3
4a(1 a)2(1 e)3,(5)118
respectively, where
As
represents the area swept by the rotor. Note that for a HAWT,
As
119
is equal to the rotor/disc area. For a CKPS, however,
As
is the annular area in the sky,
120
traversed by the kite.121
It should be noted that the total power extracted from flow, that is the sum of the power
122
harvested by the rotor and that due to reeling-out, may be obtained by calculating the
123
2
This force is called axial as it is along the axis of rotation of the rotor, which is normal to the disc/rotor
plane.
6
time-rate change of the kinetic energy of flow, that is124
Ptot =dK
dt =1
2˙mv2
1
2˙mv2
w,(6)125
where
˙m
is the mass flow rate through the actuator disc,
K
is the flow kinetic energy, and
t126
is time.127
Using the expression for the wake velocity (see equation (2)), equation (6) may be
128
simplified as129
Ptot =T v1a(1 e),(7)130
in which T= ˙m(vvw) has also been utilized.131
Equation (7) may be rewritten as132
Ptot =T ve+ (1 a)(1 e)=T vd+T vr,(8)133
where the first term on the r.h.s. is the power due to reeling-out, and the second one is the
134
power harvested by the rotor (refer to equation (5)).135
One may realize that equations (4) and (5) can be obtained from the classical actuator
136
disc theory – for a stationary disc. If the expression for the thrust of a stationary disc is
137
scaled by a factor of (1
e
)
2
and that for the power by a factor of (1
e
)
3
, then equations
138
(4) and (5) are obtained. In other words, in the classical expressions for
T
and
P
,
v
should
139
simply be replaced by
vvd
= (1
e
)
v
, i.e. the relative velocity between the disc and
140
freestream velocity. However, one should note the difference between the definition of the
141
induction factor for a stationary disc, i.e.
vrv
(1
a
), and that for a moving disc, given
142
in equation (1).143
7
3. Lift mode power generation144
Using the extended actuator disc and the blade element momentum theories, and neglecting
145
some high-order terms with negligible contributions, the induction factor,
a
, for a lift mode
146
CKPS in the straight downwind configuration may be written as (refer to Kheiri et al.,2018)
147
a
1a=1
4(Ak
As
)CL(CL
CD
)2,(9)148
where
Ak
and
As
are, respectively, the planform area of the kite and the area (e.g. annulus)
149
swept by the kite in the sky;
CL
and
CD
are the kite aerodynamic lift and drag coefficients,
150
respectively. In general,
CD
also includes the normalized or equivalent drag coefficient of the
151
tether, CDt, in addition to the drag coefficient of the kite, CDk, i.e. CD=CDk+CDt.152
In equation (9), the area ratio
σ
=
Ak/As
may, in fact, be called the solidity factor of
153
the kite system, in accordance with the terminology used for wind turbines – total blade
154
area divided by the rotor disc area (see Burton et al.,2011). It can easily be concluded from
155
the equation that, increasing either the solidity factor,
CL
and/or (
CL/CD
) will increase the
156
induction factor. It should be noted that this equation is valid only for small angles of attack,
157
very large values of the lift-to-drag ratio, i.e. (
CL/CD
)
>>
1, and a straight downwind
158
configuration. It is recalled that the straight downwind configuration corresponds to a system
159
with the tether aligned with the wind and the kite sweeping an annulus perpendicular to the
160
wind flow direction.161
In addition,
χ
=
CL
(
CL/CD
)
2
may be called the aerodynamic efficiency. As discussed
162
at the end of this section, a higher aerodynamic efficiency enhances the power generation
163
efficiency. It is also interesting to note that for an aircraft in the steady level flight, minimum
164
power required occurs when χ1/2= (C3/2
L/CD) is maximum (refer to Anderson,1999).165
On the other hand, it can be shown that there is a correlation between the lift-to-drag
166
8
ratio, (
CL/CD
), or alternatively, the aerodynamic efficiency, and the tip-speed-ratio of the kite
167
system,
λ
, which may be defined as the ratio of the crosswind speed
vc
to the undisturbed
168
relative wind velocity, v(1 e), that is169
λ=vc
v(1 e)=4(CL/CD)
4 + σCL(CL/CD)2=χ1/2
C1/2
L(1 + 1
4σχ),(10)170
where it is recalled that
e
is the ratio of reel-out speed,
vd
, to freestream velocity,
v
, i.e.
171
vd
=
ev
; also,
vc
= (
CL/CD
)(1
e
)(1
a
)
v
; refer to Kheiri et al. (2018) for details; also,
172
see Figure 4.3
173
Taking also into account the induction factor, the expression for the useful/net power
174
harvested via a kite flying steadily in lift mode during the power generation phase of a
175
pumping cycle (i.e. the cycle of reel-out or power generation phase and reel-in or power
176
consumption phase; see Figure 1) may be obtained by multiplying the thrust acting on the
177
rotor area (refer to equation (4)) by the reel-out speed, as178
PL=T vd=1
2ρAsv3
4a(1 a)(1 e)2e, (11)179
We may define a power coefficient by normalizing the useful harvested power (or power
180
output) by the wind power available to an area equal to As, as follows181
C(s)
pL=PL
1
2ρAsv3
= 4a(1 a)(1 e)2e, (12)182
where the superscript (
s
) indicates that the power has been normalized with respect to the
183
wind power available to the swept area,
As
. This is a standard definition for the power
184
coefficient for conventional wind turbines (CWTs).185
3It can easily be concluded that for σCL(CL/CD)2=σχ << 1 or a << 1, λ'(CL/CD).
9
From equation (12) and assuming that
a
and
e
are independent variables,
4
one may easily
186
find that the maximum value of C(s)
pLoccurs when e= 1/3 and a= 1/2:187
C(s)
pL,max =4
27,(13)188
which is consistent with the results obtained by De Lellis et al, (2018).189
It is well-known that for a CWT, the maximum power coefficient, according to the
190
Betz-Joukowsky limit is 16/27, which is four-times of the
C(s)
pL,max
. The power coefficient is,
191
however, only one of the performance factors. It is reminded that the premise of a CKPS is
192
to access stronger high-altitude winds when also sweeping a much larger area in the sky with
193
less material, compared to a CWT. In fact, the cumulative impact of the latter factors is
194
believed to outweigh the impact of the power coefficient, making CKPSs very appealing in
195
terms of the Levelized Cost of Energy (LCOE).196
From equation (9) and using definitions of
σ
and
χ
, in order to achieve
a
= 1
/
2 with a
197
lift mode kite, we should have198
1
4σχ = 1.(14)199
Then assuming a practical value for
χ
, say 100 (
CL
= 1
, CD
= 0
.
1), it follows that the
200
solidity factor should be 0.04 which is quite large for a CKPS, and it is, in fact, comparable
201
to the solidity factor of a typical, modern three-blade CWT (refer to Burton et al.,2011).
202
For a single kite system, this large solidity factor corresponds to a circular flight trajectory of
203
a quite small gyration radius. Flying in tight loops creates large centrifugal forces, which
204
4
In principle, the question is whether wind turbine’s power extraction capability depends on the magnitude
of the incoming wind. In reality, the magnitude of the flow velocity controls the Reynolds number which
affects aerodynamic lift and drag coefficients and in turn the power extraction capability. However, depending
on how much the incoming flow velocity is varied, the effects on the aerodynamic efficiency and thus the
power extraction capability may be negligible. For example, Laursen et al. (2007)’s CFD simulation results
for a multi-Megawatt CWT confirm that the axial induction factor changes very little when the freestream
velocity changes from 6 m/s to 10 m/s.
10
have to be sustained by the kite structure, and is also challenging from the flight control
205
point of view.206
One possible way to achieve large solidity factors while not creating large centrifugal forces
207
or hindering maneuverability and aerodynamic efficiency may be to adopt a multiple-kite
208
system concept. One such concept is what is commonly referred to as “dancing kites” or
209
“dual-airfoil”; see, for example, Payne (1976); Houska and Diehl (2007); Zanon et al. (2013);
210
Cherubini (2017). The dancing kites or the dual-airfoil concept was originally put forward to
211
reduce the losses due to the tether drag. In this concept, the individual kites are connected
212
via shorter secondary tethers flying at high speed to the longer main tether which remains
213
nearly stationary. Nevertheless, the need for a carefully designed flight control system and
214
trajectory planning make this concept challenging.215
Higher values of
χ
would obviously result into lower required values of
σ
– larger
As
for
216
the same
Ak
. An efficient wing with low drag and high lift is thus desirable, but note that
217
CDalso includes the equivalent tether drag coefficient which may not be easily reduced.218
An alternative, and also practically more satisfactory, definition of power coefficient for
219
CKPSs is proposed, by normalizing the power output by the wind power available to an
220
airborne area equal to the kite planform area,
Ak
. This definition is similar to the “Power
221
Harvesting Factor
ζ
” put forward by Diehl (2013). It is argued that based on the crosswind
222
kite power principle (refer to Loyd,1980), power is harvested by heavily loading the kite
223
aerodynamically, where the kite planform area plays an important role. In contrast, for
224
CWTs, the loading and power generated correlate to the rotor diameter and thus to the swept
225
area. This may be clarified further by comparing the axial load/thrust per unit lifting surface
226
(kite or blade) area for a typical lift mode CKPS with that for a typical CWT. Alternatively,
227
one may compare the thrust coefficients obtained by normalizing the actual axial load by the
228
11
force due to freestream dynamic pressure, as (refer to equation (4))229
C(k)
T=T
1
2ρAkv2
=4
σa(1 a)(1 e)2,(15)230
in which the superscript (
k
) indicates that the axial load/thrust has been normalized with
231
respect to the lifting surface area –
Ak
is used here to refer to both the kite and blade
232
planform area.233
For a typical CWT with
e
= 0,
σ
= 0
.
04 and
a
= 1
/
3 (to get the maximum power),
234
equation (15) yields C(k)
TCW T 22.2.235
In order to obtain the thrust coefficient value for a typical lift mode CKPS, equation (15)
236
should be simplified further via equation (9), as237
C(k)
T(σ, χ, e) = χ
(1 + 1
4χσ)2(1 e)2.(16)238
As it will be discussed in the following paragraphs, the peak value of the kite-area-normalized
239
power coefficient for a CKPS is achieved when the solidity factor is very small, or essentially
240
zero. Thus, for a typical lift mode CKPS with
e
= 1
/
3,
σ
= 0 and
χ
= 100 (where the
241
power output will be the same as that from the CWT considered above), equation (16) yields
242
C(k)
TCK P S
44
.
4, which is exactly 2 times
C(k)
TCW T
. In addition, if the CWT has three blades
243
with the total planform area equal to that of the kite (in order the CWT and the kite to
244
have the same airborne area), then the axial load acting on the kite will be 3
×
2 = 6 times
245
the load on each of the blades of the CWT.246
Therefore, it appears reasonable to define the useful power coefficient for a lift mode
247
CKPS as follows (refer to equation (11)),248
C(k)
pL=PL
1
2ρAkv3
=4
σa(1 a)(1 e)2e. (17)249
12
In other words,
C(k)
pL
determines how much net power can be harvested in lift mode per unit
250
of planform area of the airborne kite.251
It is instructive to obtain C(k)
pLas a function of independent variables σ,χ, and e. Using252
equation (9),
a
can be written in terms of
σ
and
χ
; thus, equation (17) may be re-written as
253
C(k)
pL=C(k)
pL(σ, χ, e) = χ
(1 + 1
4χσ)2e(1 e)2,(18)254
which confirms that the maximum value of
C(k)
pL
, i.e.
C(k)
pL,max
= (4
/
27)
χ
, occurs when
e
= 1
/
3
255
and
σ
= 0, meaning that to generate the maximum power, a lift mode kite with a finite
256
planform area should spin in an infinitely large area in the sky while it is reeled-out at 1
/
3
257
of the undisturbed wind speed. This conclusion is consistent with Loyd (1980)’s and Diehl
258
(2013)’s observations.259
Figure 5shows the variation of
C(k)
pL
as a function of
σ
and
χ
for
e
= 1
/
3. As seen,
C(k)
pL
260
increases as
χ
is increased and
σ
is decreased – the optimum point lies on the
σ
= 0 line. In
261
this figure and all subsequent figures for the lift mode system, the black polyline separates
262
acceptable solution points (i.e. inner region) from unacceptable solution points (i.e. outer
263
region). The acceptable region was obtained by considering
aL
1
/
2, where the subscript
264
L
is used to indicate the induction factor for a lift mode CKPS. As previously explained in
265
Section 2, this range is found by considering the fact that the relative velocity of the wake
266
flow should not be negative; please refer to equation (3).267
Figure 6shows the variation of the induction factor as a function of
σ
and
χ
for
e
= 1
/
3.
268
As expected, by increasing the solidity factor and the aerodynamic efficiency (which may be
269
linked to the tip-speed-ratio using equation (10)), the induction factor increases.270
On the other hand, it is observed that some power is spent to drag the kite at high-speed
271
in the crosswind direction. This is, in fact, the wasted portion of the total power extracted
272
from wind flow. The expression for the power loss is obtained by multiplying the component
273
13
of the drag force in the crosswind direction by the crosswind speed (see Figure 4), as follows
274
PL,loss =1
2ρAkV2
relCDcos αvc,(19)275
where Vrel is the relative fluid-kite velocity, and αis the angle of attack.276
Assuming
Vrel 'vc
, and
cos α'
1 and after some mathematical operation, the coefficient
277
of the power loss may be obtained as278
C(k)
pL,loss =C(k)
pL,loss (σ, χ, e) = χ
(1 + 1
4χσ)3(1 e)3.(20)279
It is evident from equation (20) that
C(k)
pL,loss
(
σ
= 0
, χ, e
= 1
/
3) = (8
/
27)
χ
= 2
C(k)
pL,max
, meaning
280
that two times of the useful power will be wasted (in form of drag losses) at the optimal
281
point of the lift mode power generation. This is a surprising result, but as also mentioned by
282
Diehl (2013), it may be explained by considering the fact that the optimal point for useful
283
power does not necessarily coincide with that for efficiency – if we define the efficiency as the
284
ratio of the useful power to the useful+wasted power, i.e.
η
=
e/e
+ (1
e
)(1 + 1
/
4
χσ
)
1
;
285
refer to equations (18) and (20). As seen from the equation, by increasing χ,ηis increased,286
and it follows that a 100% efficiency is achieved in the limit of
χ→ ∞
, meaning that the
287
kite should spin at infinitely high speed. However, spinning at such speeds yields zero useful
288
power (refer to equation (18)) – the rotor functions like a perfect blockage. Figure 7shows
289
variation of
C(k)
pL,loss
as a function of the solidity factor
σ
and the aerodynamic efficiency
χ
.
290
It can easily be verified that for non-zero
σ
and
χ
, and for
e
= 1
/
3, the ratio between the
291
power loss and the useful power coefficients would be less than 2.292
One interesting observation is that the amount of wasted power in lift mode is exactly the
293
same as the power harvested by the rotor (see equation (5)) – the proof is left to the reader.
294
In other words, from the total power harvested by the kite system in lift mode, the power
295
14
due to reeling-out is the useful part and that extracted by the rotor is the wasted part.296
4. Drag mode power generation297
The induction factor for a drag mode CKPS, assuming that the drag due to on-board
298
turbines acts in the same direction as the kite drag, may be written as (refer to Kheiri et al.,
299
2018)300
a
1a=1
4(Ak
As
)CL(CL
CD
)2(1
1 + κ)2,(21)301
where
κ
is the ratio of the drag/thrust of turbines to the kite drag, i.e.
κ
= (
Dp/D
) =
302
(
CDp/CD
),
Dp
and
CDp
being, respectively, the drag due to the turbines and the corresponding
303
coefficient.304
It can be shown that the same expression as in equation (21) is obtained if the drag due
305
to the turbines acts in the crosswind direction, subject to the key assumptions made in the
306
present analysis. That configuration, in fact, mimics the mechanism by which a conventional
307
wind turbine produces power. It is evident from equation (21) that the induction factor of a
308
drag mode kite, in addition to the solidity factor,
σ
= (
Ak/As
), and the kite aerodynamic
309
efficiency,
χ
=
CL
(
CL/CD
)
2
, is also dependent on the factor
κ
. As seen, by increasing
κ
(or
310
the thrust of turbines), the induction factor
a
will decrease. Increasing
κ
will reduce the
311
‘effective’ (i.e. kite-turbine system) aerodynamic efficiency
χ
=
χ/
(1 +
κ
)
2
which in turn
312
correlates with a higher angle of attack and a lower rotational speed (or crosswind speed). A
313
lower rotational speed means a lower tip-speed-ratio and thus a smaller induction factor.314
The expression for the useful harvested power (or power output) via a kite in drag mode
315
in its general form may be obtained by multiplying the thrust of turbines by the crosswind
316
speed; the final expression after several mathematical operation may be written as317
PD=1
2ρAkv3
CL(CL
CD
)2(1 a)3κ
(1 + κ)3.(22)318
15
Equation (21) is used to write
a
as a function of
σ
,
χ
, and
κ
; thus, the kite-area-normalized
319
power coefficient for a drag mode CKPS may be written as320
C(k)
pD(σ, χ, κ) = PD
1
2ρAkv3
=χ
(1 + κ)2+1
4χσ3κ(1 + κ)3.(23)321
It can be easily shown from equation (23) that the maximum value of
C(k)
pD
, i.e.
C(k)
pD,max
=
322
(4
/
27)
χ
, occurs when
κ
= 1
/
2 and
σ
= 0, meaning that, to get the maximum useful power in
323
drag mode, a kite with a finite planform area should spin in an infinitely large area in the
324
sky while it is equipped with a number of turbines, whose total thrust is 50% of the kite
325
system drag. As discussed in Section 3, the maximum kite-area-normalized power coefficient
326
for a lift mode CKPS was
CpL,max
= (4
/
27)
χ
, then confirming that a CKPS with a negligible
327
solidity factor would generate the same maximum power in lift and drag modes; a similar
328
conclusion is inferred from Loyd (1980).329
To obtain the maximum value of
C(k)
pD
for non-zero
σ
and
χ
, we need to find the optimal
330
κfrom ∂C (k)
pD/∂κ = 0 which yields the following cubic polynomial331
2κ33κ2+ 4Cκ +C+ 1 = 0,(24)332
where C=1
4σχ.333
The cubic equation (24) can have up to three real roots. However, it can be shown
334
mathematically that only one of the roots is positive; see Appendix A for more details.335
Figure 8shows the variation of the optimal
κ
, denoted by
κ
, as a function of
σ
and
χ
.
336
As seen from the figure, for non-zero σand χ, by increasing either σor χ,κis increased.337
Figure 9shows the variation of
C(k)
pD
as a function of
σ
and
χ
for
κ
=
κ
. Similarly to the
338
lift mode (refer to Figure 5),
C(k)
pD
increases as the solidity factor,
σ
, is decreased and the
339
aerodynamic efficiency, χ, is increased.340
16
Figure 10 shows the variation of the induction factor,
aD
, as a function of the solidity
341
factor,
σ
, and the aerodynamic efficiency,
χ
. The subscript
D
indicates the induction factor for
342
a drag mode CKPS. It is reminded that the local optimal
κ
is always used in the calculations.
343
As expected, by increasing either
σ
or
χ
, the induction factor is increased. It is found from
344
Figures 6and 10 that,
aD
is several times smaller than
aL
for the same
σ
and
χ
. However, a
345
direct comparison between numerical values of
aD
and
aL
may not be very sensible as
aL
is
346
defined in connection to the relative wind speed,
v
(1
e
), at the rotor, while
aD
is defined
347
in connection to the absolute wind speed, v.348
For drag mode CKPS, similarly to the lift mode system, some power is spent/wasted to349
drag the kite and the on-board turbines at high-speed in the crosswind direction. Recalling
350
that the kite drag is 1
times the thrust of turbines, the kite-area-normalized wasted power
351
in drag mode can easily be found by dividing C(k)
pD(refer to equation (23)) by κ:352
C(k)
pD,loss =C(k)
pD,loss (σ, χ, κ) = χ
(1 + κ)2+1
4χσ3(1 + κ)3.(25)353
It is found from equation (25) that
C(k)
pD,loss
(
σ
= 0
, χ, κ
= 1
/
2) = (8
/
27)
χ
= 2
C(k)
pD,max
. In other
354
words, twice the amount of the useful power has to be spent in order to spin the kite within
355
the annulus; the same conclusion was made for the lift mode system. Figure 11 shows the
356
variation of C(k)
pD,loss as a function of the solidity factor, σ, and the aerodynamic efficiency, χ.357
It can easily be verified that the ratio of the wasted power to the useful power for non-zero
358
solidity factors is always less than 2.359
The useful/net power generated in the drag mode can also be described in terms of the
360
swept area, As. Using equations (21) and (22) yields361
PD=1
2ρAsv3
4a(1 a)2κ
1 + κ.(26)362
17
Similarly to the lift mode, for a drag mode system, a power coefficient may be obtained by
363
normalizing PDby the wind power available to the swept area, as follows364
C(s)
PD=PD
1
2ρAsv3
= 4a(1 a)2κ
1 + κ,(27)365
in which it is reminded that the superscript (s) denotes normalization with respect to As.366
As seen from equation (27), to maximize
C(s)
pD
,
κ
should be infinitely large, i.e.
κ→ ∞
,
367
and
a
= 1
/
3. Theoretically, a very large
κ
may be achieved via an aerodynamically ideal
368
wing with essentially zero drag. The maximum
C(s)
pD
then becomes
C(s)
pD,max
= 16
/
27, which is
369
equal to the B-J limit and four times the
C(s)
pL,max
. Nevertheless, as previously suggested, the
370
swept-area-normalized power coefficient, such as
C(s)
pD
may not be practically significant for
371
CKPS although it is a key performance factor for CWTs.372
5. Discussion373
Figure 12 gives an interesting plot showing the ratio of the kite-area-normalized useful
374
power coefficient for the drag mode to that for the lift mode as a function of the solidity factor,
375
σ
, and the aerodynamic efficiency,
χ
. For all the points in the plot, it has been assumed
376
that
e
= 1
/
3 for the lift mode and
κ
=
κ
for the drag mode CKPS. As seen from the figure,
377
the drag mode generally outperforms the lift mode in terms of power generation (they are
378
equivalent only on the
σ
= 0 line), and this superiority becomes more evident as
σ
and
χ
are
379
increased. This is an important fundamental finding as, until now, the general perception
380
has been that the two modes are equivalent in terms of power generation potential, most
381
likely because the effects of induction factor has always been neglected – only
σ
= 0 line was
382
discernible.383
However, for practical reasons, spinning in an infinitely large area is not viable, and as a
384
result the solidity factor and the induction factor of a CKPS are expected to be perceptible.
385
18
The reason why the out-performance of drag mode over lift mode has been left seemingly
386
unnoticed in practice, at least so far, may be explained in part by the fact that most
387
experimental prototypes are small-scale kites flying over relatively large annuli/capture areas
388
in the sky (i.e. small solidity factors), thus leading to very small induction factors. For
389
example, using the values given in (Vander Lind,2013, Fig. 28.7) for the Wing 7 prototype
390
of Makani Power (rated average power: 20 kW, rated wind speed: 10 m/s) and assuming the
391
simplified straight downwind configuration, the solidity factor and the aerodynamic efficiency
392
of the kite are found as
σ
0
.
0016 and
χ
= 128, respectively. If it is further assumed that the
393
thrust/drag of on-board turbines is half of the kite drag (i.e.
κ
= 1
/
2), then using equation
394
(21), the induction factor for the Wing 7 prototype becomes
a
0
.
02; also,
C(k)
pD/C(k)
pL
1
.
03.
395
In contrast, as also discussed in Kheiri et al. (2018), for large-scale systems the induction
396
factor may be appreciable and its effects, if neglected, will result into significant overestimation
397
of the power output. This can in turn negatively affect cost estimation and examination of
398
economical prospects of those systems. For example, for a 5.5 MW kite system with the
399
solidity factor
σ
= 0
.
0048, it was shown analytically as well as computationally that the
400
induction factor was of the order of 10% which if neglected, it would result into over 20%
401
power output overestimation, which is quite significant.402
After all, Figure 12 does not demonstrate the total superiority of the drag mode over the
403
lift mode power generation. In order to draw such conclusions, it is essential to include the
404
estimation of costs incurred by the two modes of power generation, as well as to use a more
405
refined theory which takes into account, e.g., the tether drag, the wind speed variation versus
406
altitude, non-ideal inclined configurations of CKPSs, and the pumping cycle of the lift mode
407
power generation. As an example, tethers for on-board or drag mode generation concepts,
408
compared to those used for ground-based or lift mode generation, are generally thicker –
409
which means higher drag force – and heavier as they contain conductive wires to transmit
410
19
electricity to the ground station (Dunker,2018). Thicker and heavier tethers reduce power
411
output and increase the cost. In addition to heavier tethers, on-board generation systems
412
carry extra mass due to turbines and voltage converters. In fact, what rules in the end is
413
the levelized cost of energy and not the power generation potential. Obtaining the levelized
414
cost of energy for the two modes is, however, beyond the scope of the present paper and is
415
deferred to a future publication.416
We close this section by highlighting the difference between two design optimization
417
philosophies that may be adopted for CKPSs and more generally for AWE systems. See
418
Figure 13 which shows the variation of the kite-area-normalized useful power coefficient
419
divided by the aerodynamic efficiency, (
C(k)
pL
), and the swept-area-normalized useful power
420
coefficient,
C(s)
pL
, for a lift mode CKPS as a function of the induction factor,
aL
. It is assumed
421
that the reel-out speed ratio is 1/3. It is further assumed that the system initially has the
422
solidity factor
σ0
, the aerodynamic efficiency
χ0
and the corresponding induction factor
aL0
.
423
This design point is marked on the kite- and swept-area-normalized power coefficient curves
424
by points (1) and (2), respectively.425
As seen from the figure, it is possible to reach the local maximum power coefficient, by
426
reducing the induction factor to essentially zero on the
aL
(
C(k)
pL
) curve, or by increasing
427
the induction factor to 1/2 on the
aLC(s)
pL
curve. Assuming that the aerodynamic efficiency
428
remains constant, the former may be achieved by decreasing the solidity factor via expanding
429
the swept area, while the latter may be reached by increasing the solidity factor via enlarging
430
the kite planform area. In other words, in the first scenario, the optimization philosophy
431
is to keep the kite area constant and to increase the swept area, while that in the second
432
situation is to keep the swept area constant and to enlarge the kite area. From arguments
433
in Section 3, and also recalling that the premise of the AWE is to harvest high-altitude
434
winds, it appears more reasonable that the first philosophy be adopted by AWE systems
435
20
designers, while the second one by the CWT design community. It should be noted that
436
in both scenarios, a stronger tether is required to sustain higher tensions produced at the
437
local maxima. In addition, the first scenario incurs more lengths of tether, given that the
438
tether elevation angle remains constant, to allow for a larger swept area. This would in turn
439
add more mass to the system and would increase the total drag. On the other hand, in the
440
second scenario a larger kite is required, meaning larger airborne mass.441
Similarly, Figure 14 shows the curves for useful power coefficient (both the kite-area-
442
normalized divided by the aerodynamic efficiency and the swept-area-normalized) versus
443
induction factor for a drag mode CKPS. For this mode also, as seen from the figure, the
444
optimal points for the two curves do not coincide. They occur at
aD
= 0 and
aD
= 1
/
3
445
with the power coefficient values of 4
/
27 and 16
/
27, respectively (cf. the optimal points
446
for lift mode CKPS). It is also interesting to see that both
C(k)
pD
and
C(s)
pD
vary almost
447
linearly with
aD
(cf. Figure 13). As seen from the figure, to maximize the power output
448
coefficient for a drag mode kite with the initial induction factor
aD0
, one may adopt one of
449
the following two approaches: (1) decreasing
aD
essentially to zero by expanding the swept
450
area, (2) increasing
aD
to 1
/
3 by increasing the kite planform area. As discussed above, the
451
first approach seems to be favoured by the AWE community, while the second approach
452
sounds to be aligned with the design philosophy of CWTs. Finally, it should be emphasized
453
that comparing directly the optimal value of the kite-area-normalized power coefficient with
454
that of the swept-area-normalized power coefficient is meaningless (as they are two different
455
normalized forms of power output), and thus one being higher than the other does not mean
456
the corresponding design optimization philosophy has to be favoured.457
21
6. Concluding Remarks458
In this paper, expressions for useful (or usable) and wasted portions of power extracted
459
by lift and drag mode crosswind kite power systems, taking into account the induction
460
factor, were presented. These expressions had been obtained using the extended actuator disc
461
and blade element momentum theories. Two possible descriptions of the power coefficient,
462
namely the kite-area-normalized and swept-area-normalized, were given. It was discussed
463
that, in contrast to conventional wind turbines, for crosswind kite power systems, the swept-
464
area-normalized power coefficient does not properly quantifies the aerodynamic performance.
465
Alternatively, the kite-area-normalized power coefficient was proposed to be used for CKPSs.
466
The power coefficient for a lift mode CKPS was written as a function of the solidity factor,
467
aerodynamic efficiency, and the reel-out ratio, while that for a drag mode system was given as
468
a function of the first two variables plus the ratio of the on-board turbines thrust to kite drag.
469
It was shown that, for both modes and for a given value of the aerodynamic efficiency, the
470
maximum useful power is achieved when the solidity factor or essentially the induction factor
471
is zero – this is equivalent to a kite of a finite planform area spinning in an infinitely large
472
area. In other words, the kite-area-normalized power coefficient is increased by increasing
473
the aerodynamic efficiency and by decreasing the solidity factor. Moreover, it was confirmed
474
that the lift and drag mode CKPSs can harvest the same amount of useful power when the475
solidity factor (or the induction factor) is negligibly small, where also two times of the useful
476
power is spent (or wasted) to spin the kite. For non-zero induction factors, however, it was
477
found that the drag mode outperforms the lift mode in terms of potential power output.
478
Nevertheless, what matters in the end is the levelized cost of energy which can be calculated
479
more accurately using the expressions presented in this paper.480
22
Acknowledgments481
The financial support from New Leaf in the course of this research is sincerely appreciated.
482
The first author is also grateful to the Gina Cody School of Engineering and Computer Science
483
of Concordia University for a start-up research grant and to the Office of Vice-President,
484
Research and Graduate Studies for an Individual Seed Program funding.485
23
Appendix A: Roots of the cubic equation used for obtaining the optimal value486
of the ratio of turbines thrust to kite drag – equation (24)487
The cubic equation for finding the optimal value of
κ
has only one positive real root. This
488
can be shown mathematically, at least, for cases where C << 1.489
We can re-write equation (24) as follows:490
(κr)(α1κ2α2κα3) = 0,(A.1)491
where α1= 2, α2= 3 + 2r,α3=r(3 + 2r)4C, and κ1=r > 0 is the only positive root.492
The roots of the quadratic term in equation (A.1) are obtained as493
κ2,3=α2±
2α1
,(A.2)494
in which ∆ = α2
282+ 32C.495
Since
r >
0 and
C <<
1, then ∆
< α2
2
and in turn
< α2
. This will lead to
496
α2>
0 and also
α2
+
>
0. As a result, from equation (A.2) and since
α1>
0 both
497
κ2
and
κ3
will be negative. Figure 15 shows the variation of the three roots of the cubic
498
equation, obtained numerically, for a wide range of values of C.499
24
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27
wind ow
kite
tether
drum and
generator ight trajectory
to the grid
(a)
wind ow
ight trajectory
drum and
motor
tether
kite
from the grid
(b)
Figure 1: Schematic view of the lift mode or ground-based generation: (a) reel-out (or power generation)
phase, (b) reel-in (or power consumption) phase. The present drawing was inspired by the graphic design
made by Daidalos Capital (2016).
28
wind ow
ight trajectory
kite
tether
to the grid
drum
on-borad turbines
and generators
Figure 2: Schematic view of the drag mode or on-board generation. The present drawing was inspired by the
graphic design made by Daidalos Capital (2016).
29
v
vd
vd
vd
vd
vvd
vrvwvd
inlet
streamtube outlet
1
Figure 3: Schematic showing a wind turbine rotor subjected to a uniform wind field of (absolute) velocity
v
and moving downwind at constant speed
vd
=
ev
, where 0
e
1. The dashed lines show a streamtube
around the rotor, which extends from far upstream to far downstream of the rotor. The flow velocity at the
inlet of the streamtube is
vvd
and that at the outlet is
vwvd
. Also, the flow velocity at the rotor is
represented by vr.
30
x
y
va
vc
Vrel
L
D
F
φ
φγ
θ
α
1
Figure 4: Velocity vectors and aerodynamic forces acting on the airfoil section of a crosswind kite:
va
,
vc
,
and
Vrel
represent, respectively, the axial, lateral, and total relative fluid-kite velocities;
L
,
D
, and
F
are,
respectively, the lift, drag, and total aerodynamic forces;
α
is the angle of attack,
θ
is the pitch angle,
γ
is
the angle between Fand the x-axis, and φ=θ+αis the angle between Vrel and the y-axis.
31
0
0.01
0.02
0.03
0
50
100
150
200
0
5
10
15
20
25
30
0
5
10
15
20
25
σ=Ak
As
χ=CL(CL
CD)2
C(k)
pL
1
Figure 5: The kite-area-normalized coefficient of the useful power (or power output) for a lift mode CKPS,
C(k)
pL
=
PL/
(1
/
2)
ρAkv3
, as a function of the solidity factor,
σ
, and the aerodynamic efficiency,
χ
, for reel-out
ratio
e
= 1
/
3. The black polyline shows the border between acceptable (inner) and unacceptable (outer)
regions of the plot. The acceptable region is obtained by satisfying a non-negative relative wake velocity
condition, i.e. aL1/2; refer to equation (3).
32
0
0.01
0.02
0.03
0
50
100
150
200
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
σ=Ak
As
χ=CL(CL
CD)2
aL
1
Figure 6: The induction factor for a lift mode CKPS as a function of the solidity factor,
σ
, and the aerodynamic
efficiency,
χ
, for reel-out ratio
e
= 1
/
3. The black polyline shows the border between acceptable (inner) and
unacceptable (outer) regions of the plot. The acceptable region is obtained by satisfying a non-negative
relative wake velocity condition, i.e. aL1/2; refer to equation (3).
33
0
0.01
0.02
0.03
0
50
100
150
200
0
10
20
30
40
50
60
0
10
20
30
40
50
σ=Ak
As
χ=CL(CL
CD)2
C(k)
pL,loss
1
Figure 7: The kite-area-normalized coefficient of the wasted power for a lift mode CKPS,
C(k)
pL,loss
=
P(k)
L,loss/
(1
/
2)
ρAkv3
, as a function of the solidity factor,
σ
, and the aerodynamic efficiency,
χ
, for reel-out ratio
e
= 1
/
3. The black polyline shows the border between acceptable (inner) and unacceptable (outer) regions of
the plot. The acceptable region is obtained by satisfying a non-negative relative wake velocity condition, i.e.
aL1/2; refer to equation (3).
34
0
0.01
0.02
0.03
0
50
100
150
200
0.4
0.6
0.8
1
1.2
1.4
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
σ=Ak
As
χ=CL(CL
CD)2
κ
1
Figure 8: The optimal value of the ratio of on-board turbines thrust to kite drag,
κ
, for a drag mode CKPS
as a function of the solidity factor, σ, and the aerodynamic efficiency, χ.
35
0
0.01
0.02
0.03
0
50
100
150
200
0
5
10
15
20
25
30
0
5
10
15
20
25
σ=Ak
As
χ=CL(CL
CD)2
C(k)
pD
1
Figure 9: The kite-area-normalized coefficient of the useful power for a drag mode CKPS,
C(k)
pD
=
PD/
(1
/
2)
ρAkv3
, as a function of the solidity factor,
σ
, and the aerodynamic efficiency,
χ
, for optimal values of
the ratio of the on-board turbines thrust to the kite drag, κ.
36
0
0.01
0.02
0.03
0
50
100
150
200
0
0.05
0.1
0.15
0.2
0.25
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
σ=Ak
As
χ=CL(CL
CD)2
aD
1
Figure 10: The induction factor for a drag mode CKPS as a function of the solidity factor,
σ
, and the
aerodynamic efficiency,
χ
, for optimal values of the ratio of the on-board turbines thrust to the kite drag,
κ
. All the points in the plot are within the acceptable range
aD
0
.
5 which is obtained by satisfying a
non-negative wake velocity condition; refer to equation (3).
37
0
0.01
0.02
0.03
0
50
100
150
200
0
20
40
60
0
10
20
30
40
50
σ=Ak
As
χ=CL(CL
CD)2
C(k)
pD,loss
1
Figure 11: The kite-area-normalized coefficient of the wasted power for a drag mode CKPS,
C(k)
pD,loss
=
P(k)
D,loss/
(1
/
2)
ρAkv3
, as a function of the solidity factor,
σ
, and the aerodynamic efficiency,
χ
, for optimal values of
the ratio of the on-board turbines thrust to the kite drag, κ.
38
0
0.01
0.02
0.03
0
50
100
150
200
1
1.5
2
2.5
1
1.2
1.4
1.6
1.8
2
σ=Ak
As
χ=CL(CL
CD)2
C(k)
pD
C(k)
pL
1
Figure 12: The ratio of the kite-area-normalized useful power coefficient for a drag mode CKPS to that of a lift
mode CKPS,
C(k)
pD/C(k)
pL
, as a function of the solidity factor,
σ
, and the aerodynamic efficiency,
χ
, for optimal
values of the ratio of the on-board turbines thrust to the kite drag,
κ
, for the drag mode system and reel-out
ratio
e
= 1
/
3 for the lift mode system. The black polyline shows the border between acceptable (inner) and
unacceptable (outer) regions of the plot. The acceptable region is obtained by satisfying a non-negative
relative wake velocity condition, i.e. aL1/2; refer to equation (3).
39
Figure 13: Variation of the kite-area-normalized useful power coefficient divided by the aerodynamic efficiency,
(
C(k)
pL
) (red line), and the swept-area-normalized useful power coefficient,
C(s)
pL
(blue line), for a lift mode
CKPS as a function of the induction factor,
aL
, for
e
= 1
/
3. The maximum of the (
C(k)
pL
) and
C(s)
pL
curves
are located at aL= 0 and aL= 1/2, respectively.
40
Figure 14: Variation of the kite-area-normalized useful power coefficient divided by the aerodynamic efficiency,
(
C(k)
pD
) (red line), and the swept-area-normalized useful power coefficient,
C(s)
pD
(blue line), for a drag mode
CKPS as a function of the induction factor,
aD
, for optimal values of
κ
. The maximum of the (
C(k)
pD
) and
C(s)
pDcurves are located at aD= 0 and aD= 1/3, respectively.
41
0 0.5 1 1.5 2
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
C
κ
1
Figure 15: Roots of the cubic equation (24) as a function of C.
42
... Schematic drawings of (a,b) ground-based (or lift mode or pumping mode), and (c) on-board (or drag mode), power generation via CKPSs. Subfigure (a) shows the reel-out or power generation phase, and subfigure (b) shows the reel-in or power consumption phase (adapted from [6]). ...
... However, several researchers have attempted to include such effects in the aerodynamic modelling of CKPSs. Some examples are the works of [6][7][8][9][10][11][12][13]. ...
... The kite is reeling-out at one-third of the wind speed (to reach maximum power output according to Loyd's formulation), so that the relative velocity (i.e., v rel = (1 − 1/3) × 12.5 = 8.33 m/s) becomes exactly the same as the inlet flow velocity considered in the CFD simulations. It is further assumed that the kite would fly such that the aerodynamic efficiency, defined as χ = C L (C L /C D ) 2 [6], reaches the optimal value of χ = 1.23 × (1.23/0.1074) 2 = 161.3 and that there is no induction. ...
Article
Full-text available
This paper presents some results from a computational fluid dynamics (CFD) model of a multi-megawatt crosswind kite spinning on a circular path in a straight downwind configuration. The unsteady Reynolds averaged Navier-Stokes equations closed by the k−ω SST turbulence model are solved in the three-dimensional space using ANSYS Fluent. The flow behaviour is examined at the rotation plane, and the overall (or global) induction factor is obtained by getting the weighted average of induction factors on multiple annuli over the swept area. The wake flow behaviour is also discussed in some details using velocity and pressure contour plots. In addition to the CFD model, an analytical model for calculating the average flow velocity and radii of the annular wake downstream of the kite is developed. The model is formulated based on the widely-used Jensen’s model which was developed for conventional wind turbines, and thus has a simple form. Expressions for the dimensionless wake flow velocity and wake radii are obtained by assuming self-similarity of flow velocity and linear wake expansion. Comparisons are made between numerical results from the analytical model and those from the CFD simulation. The level of agreement was found to be reasonably good. Such computational and analytical models are indispensable for kite farm layout design and optimization, where aerodynamic interactions between kites should be considered.
... Since the power production potential of the crosswind kites strongly depends on the wind speed at the location of the kite, it is important to know the kite's self-induction. To the authors' knowledge, all previous engineering-type induction estimates have effectively been based on momentum-theory-based models, for instance [6][7][8][9][10][11]. References [6][7][8][9][10] used momentum-based modelling directly, whereas [11] applied an engineering approximation to vortex cylinders [12]. ...
... Since the power production potential of the crosswind kites strongly depends on the wind speed at the location of the kite, it is important to know the kite's self-induction. To the authors' knowledge, all previous engineering-type induction estimates have effectively been based on momentum-theory-based models, for instance [6][7][8][9][10][11]. References [6][7][8][9][10] used momentum-based modelling directly, whereas [11] applied an engineering approximation to vortex cylinders [12]. In the steady case and for high kite speed ratios -corresponding to a high ratio between kite eigenvelocity and onset inflow velocity -the latter yield results identical to momentum theory [5,11]. ...
Article
Full-text available
The present paper introduces a new, physically consistent definition of effective induction that should be used in engineering models for power kite performance that use aerodynamic coefficients for the wing. It is argued that in such cases it is physically inconsistent to use disc-based induction models – like momentum models – and thus a new, physically consistent induction model using vortex theory methodology is derived. Simulation results using the new induction model are compared to the previously often used momentum method and Actuator Line (AL) CFD simulations. The comparison shows that the new vortex based model is in much better agreement with the AL results than the momentum method. The new model is as computationally light as the momentum induction method.
... Various designs have been considered since the conception of AWE technology [2], including both lift and drag based power generation [3,4], soft [5,6] and rigid [7] "kite" structures, lighter- [8,9] and heavier-than-air designs, single and multi-kite [10,11] implementations, as well as other unconventional designs [12][13][14]. De Lellis, Reginatto, Saraiva, and Trofino found that the theoretical efficiency limit of AWE systems matches that of conventional wind turbines [3]. ...
... De Lellis, Reginatto, Saraiva, and Trofino found that the theoretical efficiency limit of AWE systems matches that of conventional wind turbines [3]. Kheiri, Nasrabad, and Bourgault compared lift-and drag-based systems and determined that drag-based systems have the potential to produce greater amounts of power than lift-based systems [4]. Looking specifically at soft kite technology, Folkersma, Schmehl, and Viré found that delaying separation or tripping turbulence over the kite at different phases of flight could improve the overall system performance [5]. ...
... Reaching high lift coecients is important to maximize the performance. High-delity models and ight simulators 7,8,9,10,11,12 also play an important role during the design and optimization phases of the machines. Pioneering aerodynamic models 13,14,15 have been followed by studies with viscousinviscid interaction methods 16 , Reynolds Averaged NavierStokes simulations 17,18 , and large eddy simulations 19 . ...
Article
Full-text available
The aerodynamic characteristics of a leading edge inatable (LEI) kite and a rigid-framed delta (RFD) kite were investigated. Flight data were recorded by using an experimental setup that includes an inertial measurement unit, a GPS, a magnetometer, and a multi-hole Pitot tube onboard the kites, load cells at every tether, and a wind station that measures the velocity and heading angle of the wind. These data were used to feed a ight path reconstruction algorithm that estimated the full state vector of the kite. Since the latter includes the aerodynamic force and moment about the center of mass of the kite, quantitative information about the aerodynamic characteristics of the kites was obtained. Due to limitation of the experimental setup, the LEI kite ew most of the time in post-stall conditions, which resulted in a poor maneuverability and data acquisition. This assumption was corroborated by a particular maneuver where the lift coecient decreased from 1 to 0.4, while its angle of attack increased from 35 • to 50 •. On the contrary, abundant ight data were obtained for the RFD kite during more than ten gure-eight maneuvers. Although the angle of attack was high, between 20 • and 40 • , the kite did not reach its maximum lift coecient. High tether tensions and a good maneuverability were achieved. Statistical analysis of the behavior of the lift, drag and pitch moment coecients as a function of the angle of attack and the sideslip angle allowed to identify some basic aerodynamic parameters of the kite.
Article
This paper presents two novel semi-analytical models for predicting the aerodynamic performance of crosswind kite power systems (CKPSs), where the kite induction effects on the oncoming flow are taken into account. The blade element momentum theory forms the backbone of the models. The effects of reel-out ratio, solidity factor, rotor incidence angle, side-slip angle and tether drag are included in the formulation for the axial induction factor and power output. For simplicity, the wake rotation and the tangential induction factor are neglected. Aerodynamic model 1 is developed for predicting the reel-out power with uniform inflow assumption, and it is suitable for CKPSs with ground-based power generation. Aerodynamic model 2, on the other hand, can predict both the reel-out and torque powers with either uniform or non-uniform inflow assumption, and the model can be used for CKPSs with ground-based, or on-board power generation, or with the combination of the two. Some parametric studies have been conducted for a generic kite system with pre-defined aerodynamic efficiency parameters to highlight the effects of incidence and side-slip angles. In addition, a particular CKPS and its variants are examined to show the individual and combined effects of incidence angle, side-slip angle, tether drag and airfoil shape on the induction factor and power output.
Conference Paper
Full-text available
The development of airborne wind energy systems involves many design compromises, which encourages the use of systematic optimization approaches, that rely on simple engineering models for each component. In this paper, we present results concerning rotary-based airborne wind energy devices, that are operating like autogiros tethered to the ground. This configuration results in operating conditions at unusual rotor inclinations with respect to the incoming flow. We therefore focus on the description of the aerodynamics of the rotor, comparing blade-element momentum and free-vortex approaches, so as to determine if simple aerodynamic models are adapted for being used in initial design phases.
  • U Ahrens
  • M Diehl
Ahrens, U., Diehl, M., Schmehl, R. (Eds.), 2013. Airborne Wind Energy. Berlin: Springer.